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Joos–Weinberg Equation
In relativistic quantum mechanics and quantum field theory, the Joos–Weinberg equation is a relativistic wave equations applicable to free particles of arbitrary spin (physics), spin , an integer for bosons () or half-integer for fermions (). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The spin quantum number is usually denoted by in quantum mechanics, however in this context is more typical in the literature (see #references, references). It is named after H. Joos and Steven Weinberg, found in the early 1960s. Statement Introducing a matrix;; ; : \gamma^ symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation, NB: The convention for the four-gradient in this article is , same as the Wikipedia article. Jeffery's conventions are different: . Also Jeffery uses collects the and components of the momentum operator: . The components are not to be confused with ladde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relativistic Quantum Mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light ''c'', and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. ''Non-relativistic quantum mechanics'' refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. ''Relativistic quantum mechanics'' (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mathbb). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin- particles. In Dirac representation, the four contravariant gamma matrices are :\begin \gamma^0 &= \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-spin Theory
Higher-spin theory or higher-spin gravity is a common name for field theories that contain massless fields of spin greater than two. Usually, the spectrum of such theories contains the graviton as a massless spin-two field, which explains the second name. Massless fields are gauge fields and the theories should be (almost) completely fixed by these higher-spin symmetries. Higher-spin theories are supposed to be consistent quantum theories and, for this reason, to give examples of quantum gravity. Most of the interest in the topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher-spin theories to weakly coupled conformal field theories. It is important to note that only certain parts of these theories are known at present (in particular, standard action principles are not known) and not many examples have been worked out in detail except some specific toy models (such as the higher-spin extension of pure Chern–Simons, Jackiw–Teitelboim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bargmann–Wigner Equations
:''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non-zero mass and arbitrary spin , an integer for bosons () or half-integer for fermions (). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. They are named after Valentine Bargmann and Eugene Wigner. History Paul Dirac first published the Dirac equation in 1928, and later (1936) extended it to particles of any half-integer spin before Fierz and Pauli subsequently found the same equations in 1939, and about a decade before Bargman, and Wigner. Eugene Wigner wrote a paper in 1937 about unitary representations of the inhomogeneous Lorentz group, or the Poincaré group. Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-dimensional Gamma Matrices
In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors. Gamma matrices also appear in generic settings in Riemannian geometry, particularly when a spin structure can be defined. Introduction Consider a space-time of dimension with the flat Minkowski metric, : \eta = \, \eta_\, = \text(+1, \dots, +1, -1, \dots, -1) ~, with p positive entries, q negative entries, p + q = d and . Set . The standard Dirac matrices correspond to taking and or . In higher (and lower) dimensions, one may define a group, the gamma group, behaving in the same fashion as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws, equations, and theories respect Lorentz symmetry: * The kinematical laws of special relativity * Maxwell's field equations in the theory of electromagnetism * The Dirac equation in the theory of the electron * The Standard Model of particle physics The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity. Basic properties The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose that \mathfrak is an n-dimensional Lie algebra. Let ''B'' be a nondegenerate bilinear form on \mathfrak that is invariant under the adjoint action of \mathfrak on itself, meaning that B(\operatorname_XY, Z) + B(Y, \operatorname_X Z) = 0 for all ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used. Definition If A is an matrix and B is a matrix, then the Kr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (physics)
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): :\mathbf: \beginx\\y\\z\end \mapsto \begin-x\\-y\\-z\end. It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Reversal Symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the future, in general, the macroscopic universe does not show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantum noninvasive measurements are predicted to violate time symmetry even in equilibrium, contrary to their classical counterparts, although this has not yet been experimentally confirmed. Time ''asymmetries'' generally are caused by one of three categories: # intrinsic to the dynamic physical law (e.g., for the weak force) # due to the initial conditions of the universe (e.g., for the second law of thermodynamics) # due to measurements (e.g., for the noninvasive measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charge Conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symmetry is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are P-symmetry (parity) and T-symmetry (time reversal). These discrete symmetries, C, P and T, are symmetries of the equations that describe the known fundamental forces of nature: electromagnetism, gravity, the strong and the weak interactions. Verifying whether some given mathematical equation correctly models nature requires giving physical interpretation not only to continuous symmetries, such as motion in time, but also to its discrete symmetries, and then determining whether nature adheres to these symmetries. Unlike the continuous symmetries, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
